Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other.
In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value:
Vx=f(x,y)
Vy=Vz=0
And the gradient of velocity is constant and perpendicular to the velocity itself:
| \partialVx | |
| \partialy |
=
| \gamma |
where
| \gamma |
| \partialVx | |
| \partialx |
=
| \partialVx | |
| \partialz |
=0
The displacement gradient tensor Γ for this deformation has only one nonzero term:
\Gamma=\begin{bmatrix}0&{
| \gamma} |
&0\ 0&0&0\ 0&0&0\end{bmatrix}
Simple shear with the rate
| \gamma |
| \gamma |
| \gamma |
\Gamma= \begin{matrix}\underbrace\begin{bmatrix}0&{
| \gamma} |
&0\ 0&0&0\ 0&0&0\end{bmatrix} \ simpleshear\end{matrix}= \begin{matrix}\underbrace\begin{bmatrix}0&{\tfrac12
| \gamma} |
&0\ {\tfrac12
| \gamma} |
&0&0\ 0&0&0\end{bmatrix}\ pureshear\end{matrix} +\begin{matrix}\underbrace\begin{bmatrix}0&{\tfrac12
| \gamma} |
&0\ {-{\tfrac12
| \gamma}} |
&0&0\ 0&0&0\end{bmatrix}\ solidrotation\end{matrix}
The mathematical model representing simple shear is a shear mapping restricted to the physical limits. It is an elementary linear transformation represented by a matrix. The model may represent laminar flow velocity at varying depths of a long channel with constant cross-section. Limited shear deformation is also used in vibration control, for instance base isolation of buildings for limiting earthquake damage.
See main article: Deformation (mechanics). In solid mechanics, a simple shear deformation is defined as an isochoric plane deformation in which there are a set of line elements with a given reference orientation that do not change length and orientation during the deformation.[1] This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.[2] [3] When rubber deforms under simple shear, its stress-strain behavior is approximately linear.[4] A rod under torsion is a practical example for a body under simple shear.[5]
If e1 is the fixed reference orientation in which line elements do not deform during the deformation and e1 − e2 is the plane of deformation, then the deformation gradient in simple shear can be expressed as
\boldsymbol{F}=\begin{bmatrix}1&\gamma&0\ 0&1&0\ 0&0&1\end{bmatrix}.
\boldsymbol{F}=\boldsymbol{1
In linear elasticity, shear stress, denoted
\tau
\gamma
\tau=\gammaG
where
G
G=
| E | |
| 2(1+\nu) |
Here
E
\nu
\tau=
| \gammaE | |
| 2(1+\nu) |